Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and 't Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, that participates in symmetry extension, with an additional SPT phase leads to a new theory with an emergent Abelian symmetry that also participates in a symmetry extension. The symmetry extension of the gauge theory is controlled by the discrete theta angle which comes from the SPT phase. We find that discrete theta angles can lead to two-group symmetry in 4d QCD with $SU(N),SU(N)/\mathbb{Z}_k$ or $SO(N)$ gauge groups as well as various 3d and 2d gauge theories.

中文翻译：

---

离散的θ角，对称性和异常

各种尺寸的量规理论通常允许离散的θ角，这些角是由于用附加的对称保护拓扑（SPT）相来量测全局对称性而产生的。我们通过将规范理论与拓扑量子场论（TQFT）耦合，讨论了全局对称性和't Hooft异常如何取决于离散的θ角。我们观察到，通过测量Abelian子群对称性（参与对称扩展）以及额外的SPT相，可以得出一种新的理论，该理论具有新兴的Abelian对称性，并且也参与了对称扩展。规范理论的对称性扩展由来自SPT相位的离散θ角控制。我们发现离散的θ角可以导致带有$ SU（N）的4d QCD中的两组对称性，